Abstract

Given alpha > 0 and f is an element of L-2(0, 1), we are interested in the equation {-(x(2 alpha)u'(x))' +u(x) = f(x) on (0,1], u(1) = 0. We prescribe appropriate (weighted) homogeneous boundary conditions at the origin and prove the existence and uniqueness of H-loc(2)(0, 1] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator Lu := -(x(2 alpha) u')' + u under those appropriate homogeneous boundary conditions. (C) 2011 Elsevier Inc. All rights reserved.